Thanks, Bob, for raising an excellent and important issue!
I’m fine with your definition of retrodiction.
On your definition of “retrocausality”, since I’m a block universe fan, I’m always pouncing on phrases like “alter the past” as meaningless: Any event in spacetime is what it is, and can’t “change” or “be altered”. (Just as you describe your breakfast, or as one might assign some electric field to a spacetime point E(x,t). It’s logically impossible for that value to “change”, for a fixed value of “t”.)
But there are things about the past that are hidden. (You don’t know what I had for breakfast, for example.) If you could make a decision that would determine what *I* have already had for breakfast, that would be retrocausal. It wouldn’t “alter” my breakfast; it would always have been what it was, but you still would have *caused* it. That’s the only meaningful block-universe sense of retrocausality. It *requires* hidden variables. Really, to avoid paradox, it requires intrinsically hidden variables, hidden from everyone. (Which makes the uncertainty principle handy… or at least some obvious interpretations of it.)
In your singlet-state example, it’s possible to show that it’s actually retrocausal, and not merely retrodictive. You are happy with a hidden variable, it seems (the original orientation of the two spins is hidden), so that piece is in place. The question is whether the future experimenter *causes* the earlier hidden spin, or merely *learns* about it.
The key tool to distinguish between these two options is what’s usually known as “statistical independence” (SI). SI is when the statistics on the possible values of the hidden variable are independent of the future measurement choices. SI is the natural assumption of most of the no-go theorems that leads to the conclusion there’s no spacetime-local account of entanglement.
But in your example, as in any retrocausal story, SI is generally violated. And if SI is violated, that’s direct evidence for retrocausality. (At the very least, if a future choice is causing a past hidden variable as I described retrocausality above, that would be a blatant and direct violation of SI.)
So why is SI violated? In your example, Alice makes a choice: she can align her measurement any way she wants. And lo-and-behold, she always infers a prior spin that is aligned with her choice of measurement (or anti-aligned). That’s a very strange initial spin distribution, always aligned with a future free choice. Any initial hidden variable distribution that obeyed SI could not accomplish this amazing match; therefore your account is retrocausal.
It’s probably not a coincidence that your account of this experiment is (modulo some details) basically the same account that both Nathan Agraman and I have posted in this very forum. You might be better off reading Nathan’s, as it’s much simpler: it’s basically what you have described here except with an Alice-Bob symmetry restored, and an acknowledgement that this account is explicitly retrocausal.
Best, Ken
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